martin

why don't you send a cs and some tests to back up your implementation.


Stef


On 30 sept. 05, at 08:03, Martin Snelgrove wrote:

> The documentation for Kernel-Numbers/Complex says that arcSin,  
> arcCos and arcTan are missing. "sqrt" is
> also handy for implementing some of these.
>
> Following are versions of these functions, plus a cut-and-paste set  
> of manual tests from a Workspace. I'm new to
> Smalltalk, so there are certainly lots more parentheses than  
> necessary, plus I'm sure lots of other style failures.
> There will also be lots of numerical and efficiency improvements  
> possible. I hope the functions provide a start, though.
>
> sqrt
>     "complex square root: principal value taken as upper half-plane"
>     | |
>     imaginary = 0.0 ifTrue:
>         [real >= 0.0 ifTrue:
>             [^real sqrt]
>                         ifFalse:
>             [^real negated sqrt * 1 i]
>         ].
>     ^ (0.5 * (self ln)) exp.
>
>
>
> arcSin
>     "analytic continuation of inverse sine function"
>     |  |
>     ^ ((self * 1 i) + (1 - (self squared)) sqrt) ln * (-1 i).
>
>
> arcCos
>     "inverse of cosine function extended to Complex"
>     | |
>     ^ (self + ((self squared - 1) sqrt)) ln * -1 i .
>
>
> arcTan
>     "inverse of tan on the complex domain"
>     | iself |
>     iself := 1 i * self.
>      ^ 0.5 i *  ((1.0 - iself) / (1.0 + iself)) ln.
>
>
> Here are some simple tests, copied & pasted from Workspace:
>
> Transcript clear.
> { 0 . 0 + 0i . 1 . 1 + 0 i . 3 + 4 i . 1.3 + 4 i . 1 i . -1 i .}  
> do: [: z |
>     Transcript
>         show: 'z = '; show: z ;
>         show: '  z sqrt ='; show: z sqrt ;
>         show: '  z arcSin = '; show: z arcSin;
>         show: '  z arcSin sin = '; show: z arcSin sin;
>         show: '  z sin arcSin = '; show: z sin arcSin;
>         show: '  z arcCos = '; show: z arcCos;
>         show: '  z arcCos cos = '; show: z arcCos cos;
>         show: '  z cos arcCos = '; show: z cos arcCos;
>         show: String crlfcrlf].
>
> And the results are:
>
> z = 0  z sqrt =0.0  z arcSin = 0.0  z arcSin sin = 0.0  z sin  
> arcSin = 0.0  z arcCos = 1.570796326794897  z arcCos cos =  
> 1.224646799147353e-16  z cos arcCos = 0.0
>
> z = 0 + 0 i  z sqrt =0.0  z arcSin = 0.0 + 0.0 i  z arcSin sin =  
> 0.0 + 0.0 i  z sin arcSin = 0.0 + 0.0 i  z arcCos =  
> 1.570796326794897 + 0.0 i  z arcCos cos = 6.12323399573677e-17 +  
> 0.0 i  z cos arcCos = 0.0 + 0.0 i
>
> z = 1  z sqrt =1.0  z arcSin = 1.570796326794897  z arcSin sin =  
> 1.0  z sin arcSin = 1.0  z arcCos = 0.0  z arcCos cos = 1.0  z cos  
> arcCos = 1.0
>
> z = 1 + 0 i  z sqrt =1.0  z arcSin = 1.570796326794897 + 0.0 i  z  
> arcSin sin = 1.0 - 6.12323399573677e-17 i  z sin arcSin = 1.0 + 0.0  
> i  z arcCos = 0.0 + 0.0 i  z arcCos cos = 1.0 + 0.0 i  z cos arcCos  
> = 1.0 + 1.110223024625156e-16 i
>
> z = 3 + 4 i  z sqrt =2.0 + 1.0 i  z arcSin = 0.633983865639184 +  
> 2.305509031243485 i  z arcSin sin = 3.000000000000056 +  
> 4.00000000000001 i  z sin arcSin = 0.1415926535897932 - 4.0 i  z  
> arcCos = 0.93681246115572 - 2.305509031243477 i  z arcCos cos = 3.0  
> + 4.0 i  z cos arcCos = 2.999999999999935 + 4.00000000000009 i
>
> z = 1.3 + 4 i  z sqrt =1.659208873072136 + 1.205393746657626 i  z  
> arcSin = 0.306199697879277 + 2.14098384538913 i  z arcSin sin =  
> 1.299999999999996 + 3.999999999999993 i  z sin arcSin =  
> 1.300000000000133 + 3.99999999999994 i  z arcCos =  
> 1.264596628915619 - 2.140983845389134 i  z arcCos cos = 1.3 + 4.0  
> i  z cos arcCos = -1.3 - 4.0 i
>
> z = 0 + 1 i  z sqrt =0.7071067811865475 + 0.7071067811865475 i  z  
> arcSin = 0.0 + 0.881373587019543 i  z arcSin sin = 0.0 + 1.0 i  z  
> sin arcSin = 0.0 + 1.0 i  z arcCos = 1.570796326794897 -  
> 0.881373587019543 i  z arcCos cos = 1.47827945580917e-16 + 1.0 i  z  
> cos arcCos = 0.0 - 1.0 i
>
> z = 0 - 1 i  z sqrt =0.7071067811865475 - 0.7071067811865475 i  z  
> arcSin = 0.0 - 0.881373587019543 i  z arcSin sin = 0.0 - 1.0 i  z  
> sin arcSin = 0.0 - 1.0 i  z arcCos = 1.570796326794897 +  
> 0.881373587019543 i  z arcCos cos = 2.536326566618168e-17 - 1.0 i   
> z cos arcCos = 0.0 - 1.0 i
>
> Note that these functions aren't one-to-one, so "z sin arcsin"  
> isn't necessarily the identity. The real part can take jumps of  
> multiples of Pi,
> signs can change, etcetera. For example, note:
> (3 + 4 i) sin arcSin                          -> 0.1415926535897932  
> - 4.0 i
> (3 + 4 i) sin arcSin sin arcSin       -> 0.1415926535897932 - 4.0 i
> The first time through, "sin arcSin" jumps branches; after that  
> it's the identity.
>
>
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